Can any one say what is the necessary and sufficient conditions for

[godel]

can anyone say what is the necessary and sufficient conditions for being a tuple of real numbers ,a vector?

 

[ehild]

You can represent a vector by its components, which are numbers, real or complex. But it is only the representation of vectors.
The definition of "vector" does not need numbers. Vector is an entity that has "magnitude" and "direction", can be added to an other vector and multiplied by a "scalar", and so on. The displacement of a body is vector, the velocity is also a vector, the electric field is vector...
The wavefunctions in quantum mechanics are considered vectors, the vibrations of a molecule also can be treated as vectors and there are other examples.

ehild